Title: Tukey theory of ultrafilters – a sampling of results and open problems

Abstract: Tukey reduction between partially ordered sets originated in Tukey’s work extending the notion of convergence to general topological spaces. When restricted to the class of ultrafilters on the natural numbers, Tukey reduction generalizes the well-studied Rudin-Keisler reduction; the Tukey equivalence classes are exactly the cofinal types of the ultrafilters. We will survey some of the results on the structure of Tukey types of ultrafilters on the natural numbers as well as more generally on Boolean algebras. The talk will include open problems, some of which are accessible to graduate students.